A fair die is rolled nine times. What is the probability that an odd number1,3 or 5 will occur less than 3 t?

Question

A fair die is rolled nine times. What is the probability that an odd number1,3 or 5 will occur less than 3 t?

Answer 1

Let X be the number of times an odd number is rolled.  X has the binomial distribution with n = 9 trials and success probability p = 0.5



In general, if X has the binomial distribution with n trials and a success probability of p then

P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

for values of x = 0, 1, 2, ..., n

P[X = x] = 0 for any other value of x.

The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.

Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

X ~ Binomial( n , p )

the mean of the binomial distribution is n * p = 4.5

the variance of the binomial distribution is n * p * (1 - p) = 2.25

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.5

The Probability Mass Function, PMF,

f(X) = P(X = x) is:

P( X =  0 ) =  0.001953125

P( X =  1 ) =  0.01757813

P( X =  2 ) =  0.0703125

P( X =  3 ) =  0.1640625

P( X =  4 ) =  0.2460938

P( X =  5 ) =  0.2460938

P( X =  6 ) =  0.1640625

P( X =  7 ) =  0.0703125

P( X =  8 ) =  0.01757813

P( X =  9 ) =  0.001953125

P(X < 3) = P( X ≤ 2 )

= P(X = 0) + P(X = 1) + P(X = 2)

=  0.08984375